Electromagnetics

==gauge invariance

the electromagnetic potential is a connection on a U(1)-bundle on spacetime whose curvature is the electromagnetic field
the electromagnetism is a gauge field theory with structure group U(1)

==Lorentz force

almost all forces in mechanics are conservative forces, those that are functions only of positions, and certainly not functions of velocities
Lorentz force is a rare example of velocity dependent force

==polarization of light

Lagrangian for a charged particle in an electromagnetic field
\(L=T-V\)
\(L(q,\dot{q})=m||\dot{q}||-e\phi+eA_{i}\dot{q}^{i}\)
action
\(S=-\frac{1}{4}\int F^{\alpha\beta}F_{\alpha\beta}\,d^{4}x\)
Euler-Lagrange equations
\(p_{i}=\frac{\partial{L}}{\partial{\dot{q}^{i}}}=m\frac{\dot{q}_{i}}{||\dot{q}_{i}||}+eA_{i}=mv_{i}+eA_{i}\)
\(F_{i}=\frac{\partial{L}}{\partial{{q}^{i}}}=\frac{\partial}{\partial{{q}^{i}}}(eA_{j}\dot{q}^{j})=e\frac{\partial{A_{j}}}{\partial{q}^{i}}\dot{q}^{j}}}\)
equation of motion
\(\dot{p}=F\) Therefore we get
\(m\frac{dv_{i}}{dt}=eF_{ij}\dot{q}^{j}\). This is what we call the Lorentz force law.
force on a particle is same as \(e\mathbf{E}+e\mathbf{v}\times \mathbf{B}\)

total energy of a charge particle in an electromagnetic field
\(E=\frac{1}{2m}(p_j-eA_{j})(p_j-eA_j)+q\phi\)
replace the momentum with the canonical momentum
- similar to covariant derivative

http://www.math.toronto.edu/~colliand/426_03/Papers03/C_Quigley.pdf
Feynman's proof of Maxwell equations and Yang's unification of electromagnetic and gravitational Aharonov–Bohm effects

==related items

ELECTROMAGNETIC THEORY AND COMPUTATION

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